User:Frostburn/Geometric algebra for regular temperaments: Difference between revisions

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 This is a work-in-progress for my (Frostburn) thoughts about using geometric algebra to work with regular temperaments. I'm trying to write this in such a way that both geometers and musicians can follow the line of thought.
 == Vals ==
-The simplest kinds of tunings only have a finite set of notes like our western [[12edo|12-tone equal temperament]]. To understand such scales harmonically we need to know how to translate musically meaningful fractions such as [[3/2]] or [[5/3]] into numbers of steps in the scale of interest. Let's start with 5 equal divisions of the octave (the octave is the fraction 2/1). How should we represent the next prime 3/1? If 5 steps is the octave then the closest we can get to 3/1 is with 8 steps. The next prime 5/1 takes 12 steps of our 5-tone scale. These step counts are collected into a vector <math>< 5, 8, 12 ]</math> known as a [[Val|val]] where the prime factorization of intervals of interest are collected into vectors known as [[Monzo|monzos]]: <math>3/2 \mapsto [ -1, 1, 0 ></math> and <math>5/3 \mapsto [ 0, -1, 1 ></math>. We can now use the dot product to work out how many steps we need to represent 3/2 (known as the fifth) <math>< 5, 8, 12 | -1, 1, 0 > = 8 - 5 = 3</math> steps of 5-tone equal temperament. Similarly 5/3 (known as the major sixth) equals <math>< 5, 8, 12 | 0, -1, 1 > = 12 - 8 = 4</math> steps.
+The simplest kinds of tunings only have a finite set of notes like our western [[12edo|12-tone equal temperament]]. To understand such scales harmonically we need to know how to translate musically meaningful fractions such as [[3/2]] or [[5/3]] into numbers of steps in the scale of interest. Let's start with 5 equal divisions of the octave (the octave is the fraction 2/1). How should we represent the next prime 3/1? If 5 steps is the octave then the closest we can get to 3/1 is with 8 steps. The next prime 5/1 takes 12 steps of our 5-tone scale. These step counts are collected into a vector <math>\langle 5, 8, 12 \rbrack</math> known as a [[Val|val]] where the prime factorization of intervals of interest are collected into vectors known as [[Monzo|monzos]]: <math>3/2 \mapsto \lbrack -1, 1, 0 \rangle</math> and <math>5/3 \mapsto \lbrack 0, -1, 1 \rangle</math>. We can now use the dot product to work out how many steps we need to represent 3/2 (known as the fifth) <math>\langle 5, 8, 12 | -1, 1, 0 \rangle = 8 - 5 = 3</math> steps of 5-tone equal temperament. Similarly 5/3 (known as the major sixth) equals <math>\langle 5, 8, 12 | 0, -1, 1 \rangle = 12 - 8 = 4</math> steps.
 === Combining vals ===
-Notice that it doesn't matter if we scale our val. <math>< 10, 16, 24 ]</math> represents only 5 unique steps within 10-tone equal temperament. There is no integral monzo (that is, no rational number) that would map to an odd number of steps in this rescaled version. Because the sizes don't matter when we add vals together we're producing a sort of average. Let's take the val for 7-tone equal temperament <math>< 7, 11, 16 ]</math> and add it to <math>< 5, 8, 12 ]</math>. The result is <math>< 12, 19, 28 ]</math> which just happens to line up with the val for our familiar 12-tone equal temperament. If we add the vals for 7-tone and 12-tone together we get <math>< 17, 27, 40 ]</math> which is different from the optimal ([[Patent val|patent]]) val for 17-tone equal temperament <math>< 17, 27, \mathbf{39} ]</math>. By insisting that the first non-zero component of a val is positive and that all of the components are in lowest terms (GCD = 1) we get 1-[[Wedgies and multivals|wedgies]] which are unique identifiers of tunings.
+Notice that it doesn't matter if we scale our val. <math>\langle 10, 16, 24 \rbrack</math> represents only 5 unique steps within 10-tone equal temperament. There is no integral monzo (that is, no rational number) that would map to an odd number of steps in this rescaled version. Because the sizes don't matter when we add vals together we're producing a sort of average. Let's take the val for 7-tone equal temperament <math>\langle 7, 11, 16 \rbrack</math> and add it to <math>\langle 5, 8, 12 \rbrack</math>. The result is <math>\langle 12, 19, 28 \rbrack</math> which just happens to line up with the val for our familiar 12-tone equal temperament. If we add the vals for 7-tone and 12-tone together we get <math>\langle 17, 27, 40 \rbrack</math> which is different from the optimal ([[Patent val|patent]]) val for 17-tone equal temperament <math>\langle 17, 27, \mathbf{39} \rbrack</math>. By insisting that the first non-zero component of a val is positive and that all of the components are in lowest terms (GCD = 1) we get 1-[[Wedgies and multivals|wedgies]] which are unique identifiers of tunings.
 === Geometric interpretation ===
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 D lines that pass through the origin find representation in the [[Wikipedia:Geometric Algebra|Geometric Algebra]] <math>\mathcal G(3,0)</math>. Our vals are now vectors, or points if you think projectively, in this space.
-:<math> < 5, 8, 12 ] \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math>
+:<math> \langle 5, 8, 12 \rbrack \mapsto 5e_1 + 8e_2 + 12e_3 =: \overleftarrow{5} </math>
 I will use the term pseudoscalar to refer to the combination of all basis vectors and denote it with ''i''.
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 With a temperament of rank 2 the period <math>\overrightarrow{p}</math> and the generator <math>\overrightarrow{g}</math> together with n - 2 commas <math>\overrightarrow{c}_i</math> that are tempered out form a basis for the whole n-dimensional JI subgroup and can thus be inverted to produce a [[mapping]] for the temperament.
-Let's choose <math>\overrightarrow{p} = \overrightarrow{2/1} = [1, 0, 0></math> and <math>\overrightarrow{g} = \overrightarrow{3/2} = [-1, 1, 0> </math> as the period and generator for Meantone. In this case there's only a single comma <math>\overrightarrow{c} = \overrightarrow{81/80} = [-4, 4, 1 ></math>. Inverting the matrix <math>[\overrightarrow{p}, \overrightarrow{g}, \overrightarrow{c}]</math> gives us the mapping:
+Any interval within the JI subgroup can be written as
+:<math>
+\overrightarrow{m} = x \overrightarrow{p} + y \overrightarrow{g} + z_1 \overrightarrow{c_1} + \ldots + z_{n-2} \overrightarrow{c_{n-2}}
+</math>
+The wedge product has some nice properties that allow us to extract <math>x</math> and <math>y</math>. Let's first calculate the relevant hyperwedge.
+:<math>\begin{align}
+\mathbf{H} &= \overrightarrow{p} \wedge \overrightarrow{g} \wedge \overrightarrow{c_1} \wedge \ldots \wedge \overrightarrow{c_{n-2}} \\
+\mathbf{H} &= \overrightarrow{p} \wedge \overrightarrow{g} \wedge \mathbf{T}i
+\end{align}</math>
+where we've used the fact that the commas span the dual of the temperament.
+Finally we have
+:<math>\begin{align}
+x &= \frac{\overrightarrow{m} \wedge \overrightarrow{g} \wedge \mathbf{T}i}{\mathbf{H}} \\
+y &= \frac{\overrightarrow{p} \wedge \overrightarrow{m} \wedge \mathbf{T}i}{\mathbf{H}}
+\end{align}</math>
+where the division is inverse scalar multiplication (or right multiplication by the geometric inverse. Same thing in this case where the relevant subspace attitudes align).
+Calculating the wedges can be computationally intensive so it's often convenient to precalculate the mapping for formal primes in the subgroup.
+Let's do that for meantone and choose <math>\overrightarrow{p} = \overrightarrow{2/1} = [1, 0, 0></math> and <math>\overrightarrow{g} = \overrightarrow{3/2} = [-1, 1, 0> </math> as the period and generator for Meantone. In this case there's only a single comma <math>\overrightarrow{c} = \overrightarrow{81/80} = [-4, 4, -1 ></math> so we can calculate the mapping in terms of that too, but in general we don't need to know the commas as long as we know the wedgie for the temperament.
+:<math>\begin{align}
+\overrightarrow{2/1} &= 1 \overrightarrow{p} + 0 \overrightarrow{g} + 0 \overrightarrow{c} \\
+\overrightarrow{3/1} &= 1 \overrightarrow{p} + 1 \overrightarrow{g} + 0 \overrightarrow{c} \\
+\overrightarrow{5/1} &= 0 \overrightarrow{p} + 4 \overrightarrow{g} - 1 \overrightarrow{c}
+\end{align}</math>
+Transposing the coefficients gives us the mapping:
 :<math>\begin{align}
 \overleftarrow{p} &= < 1, 1, 0 ] \\
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 For example <math>\overleftarrow{p} \cdot \overrightarrow{5/3} = -1</math> and <math>\overleftarrow{g} \cdot \overrightarrow{5/3} = 3</math>. Indeed <math>\frac{5}{3} \sim (\frac{2}{1})^{-1}(\frac{3}{2})^3 = \frac{27}{16}</math> in Meantone. Musically this means that the major sixth is constructed by stacking three perfect fifths and reducing by an octave. If we want to be pedantic we can calculate <math>\overleftarrow{c} \cdot \overrightarrow{5/3} = -1</math> for the final numerically correct expression <math>\frac{5}{3} = \frac{27}{16}(\frac{81}{80})^{-1}</math>.
-If you're worried about the potentially non-integral period ''e''<sub>1</sub>/''d'', I'm pretty sure the non-integral components of the matrix inverse will fall onto the comma mapping vals so they don't matter in the end. I also believe that its always possible to find a period monzo with integer components that has the same value in cents as the non-integral period, but I don't know how to find it algorithmically.
+If you're worried about the potentially non-integral period ''e''<sub>1</sub>/''d'', the fractions in the mapping cluster on the commas so they don't matter in the end. The period and generator always get integral mappings. It should also always be possible to find a period monzo with integer components that has the same value in cents as the non-integral period, but I don't know how to find it algorithmically.
 == Optimizing database keys ==
 As mentioned before wedgies can be used as unique identifiers of temperaments. The geometric algebra for n-dimensional space is 2<sup>n</sup>-dimensional, but most of the wedgie components are zero and can be dropped. Additionally in the case of rank-2 temperaments there's a neat little [[Wedgies_and_multivals#Constrained_wedgies|trick]] that allows us to drop even more components. In the subgroup ''o''.''p''<sub>2</sub>.''p''<sub>3</sub>.(…).''p''<sub>n</sub> calculate
-:<math>\mathbf{T}' = < 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) ] \wedge < 0, W_{12}, W_{13}, \ldots, W_{1n} ]</math>
+:<math>\mathbf{T}' = \langle 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) \rbrack \wedge \langle 0, W_{12}, W_{13}, \ldots, W_{1n} \rbrack </math>
 If the temperament is reasonably close to just intonation it can be recovered by rounding <math>\mathbf{T}'</math> to the nearest integer.
 Examples in 2.3.5 include [[Meantone_family#Meantone|Meantone]] ~ [1, 4], [[Augmented_family#Augmented|Augmented]] ~ [3, 0] and [[Limmic_temperaments#5-limit_.28blackwood.29|Blackwood]] ~ [0, 5], [[Slendro_clan#Semaphore|Semaphore]] ~ [2, 1] in 2.3.7 and [[Gamelismic_clan#Miracle|Miracle]] ~ [6, -7, -2] in 2.3.5.7.
+Note that this generalizes to arbitrary rank. For example in rank 3:
+:<math>\mathbf{T}' = \langle 1, \log_o(p_2), \log_o(p_3), \ldots, \log_o(p_n) \rbrack \wedge \langle \langle 0, \ldots, 0, W_{123}, W_{124}, \ldots, W_{1(n-1)n} \rbrack \rbrack</math>
+Where <math>W_{1ij}</math> are the 3-wedgie components associated with ''e<sub>1</sub>'' of a temperament <math>\mathbf{T}</math> and are back-filled into a 2-val. If <math>\mathbf{T}</math> is sufficiently good we have <math>\mathbf{T} = \mathrm{round}(\mathbf{T}')</math>.
+As an example [[Marvel_family#Marvel|Marvel]] ~ [1, 2, -2] which is just the non-two monzo components of the [[225/224|marvel comma]] reversed and with alternating signs. This makes sense: a comma's two's component can be deduced from the other components by assuming that the comma has a positive size less than the octave.
+== Improving the notation ==
+It might be possible to consistently define everything so that the Tenney weighted metric is incorporated in the basis vectors. If ''e''<sub>i</sub> are the GA basis vectors that square to 1 we can define basis vectors
+:<math>\begin{align}
+\overrightarrow{e_i} &:= e_i \log(p_i) \\
+\overleftarrow{e_i} &:= e_i / \log(p_i) \\
+\end{align}
+\implies \overleftarrow{e_i}\overrightarrow{e_i} = 1</math>
+where ''p''<sub>i</sub> are the [[Just_intonation_subgroup|formal primes]] of the fractional just intonation subgroup.
+It should be noted that the pseudoscalar relevant for tempering is <math>\overleftarrow{i} := \prod_i \overleftarrow{e_i}</math>.
+This also gives <math>\overleftarrow{JIP}</math> a particularly nice representation as the sum of ''e''<sub>i</sub>.
+I will look into it more before turning this draft into a page in the Wiki proper.
+It could also be nice to use units that make sure that all of the formulae make sense. Starting with sound frequency in Hz we identify ratios of frequencies as musically interesting such as the fifth = 660 Hz / 440 Hz. To make working with ratios easier we take the logarithm and measure pitch in cents: <math>\mathrm{fifth}_{cents} = \log_2(\frac{3}{2}) * 1200 ¢ = \log_2(3) * 1200 ¢ - \log_2(2) * 1200 ¢ = \overleftarrow{JIP} \cdot \overrightarrow{3/2}</math>. If we leave the just intonation point unit-less then monzo components will be measured in cents and val components in reciprocal cents <math>¢^{-1}</math>.